Continuous Distributions
Standard Normal Distribution
PDF:
f  z   k exp   z 2 2  ,
k
1
2
1

2
   exp   z 2   2 
k

Mean = Mode:
Variance:
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E Z   0
Var  Z   1
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Normal Distribution
PDF:
 1  x   2 
f  x |  ,    exp   
,
 2    



k
k
1
2


 1  x   2 
dx


2

   exp   


 
2



 k



EX   
Mean = Mode:
Var  X    2
Variance
Uniform
PDF:
1, 0    1
f    
0, elsewhere
Mean:
E     1 2  0.5
Var     1 12  0.08333
Variance:
PDF
1
0
0
0.5
1
pi
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Beta Distribution
  1
k  1 1   
, 0    1 
PDF: g    g  |  ,    
, k 
elsewhere 
0,
1
  ,  

    
      
1
      
1
  1
 1
   ,        1   
d 
k
    
0
E  
Mean:
Var    
Variance:

 
, ,   0

, ,   0
        1
2
3.0
3.0
2.5
   1
2.0
PDF
PDF
2.5
    0.5
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
X
0.6
0.7
0.8
0.9
1.0
X
3.0
4.0
2.5
3.5
  2
    12
3.0
PDF
2.0
PDF
0.5
1.5
2.5
2.0
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
X
0.5
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
X
12.0
10.0
PDF
PDF
3.0
  1 2,   2
8.0
6.0
4.0
  8,   3
2.0
1.0
2.0
0.0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
X
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0.2
0.3
0.4
0.5
X
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Mode:
Mode    
 1
, ,   0
  2
  x    x  1 !, x  1, 2,
  x   ( x  1)  x  1 , x  .
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JMP Script for Beta Distribution
Figure 1. Beta (2, 2) PDF and CDF.
Beta PDF and CDF
https://courseware.vt.edu/users/holtzman/common/data/JMP_Scripts/BetaPDFandCDF.J
SL
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Minitab Implementation for Bolstad Chapter 7, Example 12
Normal Approximation to the Beta PDF
Bolstad Chapter 7 Example 12
Beta(12, 25), Mean=0.3243, Var=0.005767, SD=0.07594
0.4
Variable
Beta pdf
Normal pdf
5
Y-Data
4
3
2
1
0
0.0
0.1
0.2
0.3
x
0.4
0.5
0.6
To approximate P(Y>0.4)
Normal Approximation to the Beta CDF
Bolstad Chapter 7 Example 12
Beta(12, 25), Mean=0.3243, Var=0.005767, SD=0.07594
0.4
1.0
Variable
Beta(12,25) CDF
Normal A pprox
Y-Data
0.8
0.6
0.4
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
To approximate P(Y>0.4)
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The Minitab project that calculated and graphed this distribution is on line at
http://courseware.vt.edu/users/holtzman/common/data/Examples/BetaNormalApproxBols
tadCh7Example12.MPJ
For Bolstad Chapter 7, Example 12, there is an error. The answer is not 0.3406. The
correct answer using Bolstad’s style of Normal table that gives P(0 < Z < z) is
P(Y > 0.4) = 0.5 – 0.3406 = 0.1594
Using Minitab, we find the normal approximation is
P(Y > 0.4) = 0.15942
Using Minitab, we find the exact Beta-distribution value is
P(Y > 0.4) = 0.16201
so the normal approximation was correct to two decimal places and two significant
figures.
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Gamma Distribution
g  x   g  x | r, v   kx r 1evx ,
PDF:
k  v r 
r
1
r  
 r 1  vx
   x e dx  r 
v 
k
r
E  X   , r, v  0
v
Mean:
Var  X  
Variance:
r
v2
1.0
PDF
0.8
r=v=4
0.6
0.4
0.2
0.0
0
1
2
3
4
5
X
Gamma PDF and CDF JMP script
If Y ~ Chi-Squared with d degrees of freedom, then Y ~ Gamma with r = d/2 and v = ½.
d 2
d
12
d 2
Var Y  
 2d
2
1 2 
E Y  
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Exercises
1. Using Minitab, JMP, R, SAS, or whatever you like, for Y~Beta(1, 1)
a. calculate and graph the PDF and CDF of Beta(1,1),
b. calculate the mean, median, variance, standard deviation, and mode.
2. Repeat for Beta (0.5, 0.5).
3. Repeat for Beta (0.5, 2.0).
4. Repeat for Beta (2.0, 0.5).
5. Repeat for Beta (2, 2).
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